The radius of a circle is changing at the rate of 1/π inches per second. At what rate, in square inches per second, is the circle’s area changing when the radius is 5 inches?

formula for the area A of a circle in terms of its radius r:

   A = πr2

It is important to remember that both r, and hence, A are functions of time, t.  Take the derivative of both sides with respect to t and obtain

   dA/dt = 2πr(dr/dt)

Note that we are using implicit differentiation here, since A and r are (implicit) functions of t.  Now we can find the required answers.

r = 5 inches.

Substitute r = 5 and dr/dt = 3 to get

   dA/dt = 2π5·3 = 30π ≅ 95.25

(the units are in2/min.)

r = 22 inches

   dA/dt = 2π22·3 = 132π ≅ 414.69

Hope that helps.